Well, it is normal that the contact form does not send email anymore, as it is defined on this line $wpcf7_data->skip_mail = true;
.
This code is used if you want something else than the default posting (sending in email), as described in the tutorial.
This code, however, should be placed in your theme's functions.php
file (if file exists, create it). But still, the email will not be send.
If you want to skip emails and perform other action, then leave this piece of code
$wpcf7_data->skip_mail = true;
and add your logic after this line.
Describe more precisely what do you want to do (instead of sending email) in your question.